{"id":1373,"date":"2018-02-26T20:35:47","date_gmt":"2018-02-26T20:35:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=back-matter&p=1373"},"modified":"2018-02-26T20:35:51","modified_gmt":"2018-02-26T20:35:51","slug":"mathematical-formulas","status":"publish","type":"back-matter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/back-matter\/mathematical-formulas\/","title":{"raw":"Mathematical Formulas","rendered":"Mathematical Formulas"},"content":{"raw":"

Quadratic formula<\/strong><\/p>\n

If $$ a{x}^{2}+bx+c=0, $$ then $$ x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}$$<\/p>\n
Geometry<\/span><\/caption>\n
Triangle of base $$ b $$ and height $$ h$$<\/th>\nArea $$ =\\frac{1}{2}bh$$<\/th>\n\n<\/tr><\/thead>
Circle of radius $$ r$$<\/td>\nCircumference $$ =2\\pi r$$<\/td>\nArea $$ =\\pi {r}^{2}$$<\/td>\n<\/tr>
Sphere of radius $$ r$$<\/td>\nSurface area $$ =4\\pi {r}^{2}$$<\/td>\nVolume $$ =\\frac{4}{3}\\pi {r}^{3}$$<\/td>\n<\/tr>
Cylinder of radius $$ r $$ and height $$ h$$<\/td>\nArea of curved surface $$ =2\\pi rh$$<\/td>\nVolume $$ =\\pi {r}^{2}h$$<\/td>\n<\/tr><\/tbody><\/table>

Trigonometry<\/strong><\/p>\n

Trigonometric Identities<\/em><\/p>\n

  1. $$\\text{sin}\\,\\theta =1\\text{\/}\\text{csc}\\,\\theta $$<\/li>\n
  2. $$\\text{cos}\\,\\theta =1\\text{\/}\\text{sec}\\,\\theta $$<\/li>\n
  3. $$\\text{tan}\\,\\theta =1\\text{\/}\\text{cot}\\,\\theta $$<\/li>\n
  4. $$\\text{sin}({90}^{0}-\\theta )=\\text{cos}\\,\\theta $$<\/li>\n
  5. $$\\text{cos}({90}^{0}-\\theta )=\\text{sin}\\,\\theta $$<\/li>\n
  6. $$\\text{tan}({90}^{0}-\\theta )=\\text{cot}\\,\\theta $$<\/li>\n
  7. $${\\text{sin}}^{2}\\,\\theta +{\\text{cos}}^{2}\\,\\theta =1$$<\/li>\n
  8. $${\\text{sec}}^{2}\\,\\theta -{\\text{tan}}^{2}\\,\\theta =1$$<\/li>\n
  9. $$\\text{tan}\\,\\theta =\\text{sin}\\,\\theta \\text{\/}\\text{cos}\\,\\theta $$<\/li>\n
  10. $$\\text{sin}(\\alpha \u00b1\\beta )=\\text{sin}\\,\\alpha \\,\\text{cos}\\,\\beta \u00b1\\text{cos}\\,\\alpha \\,\\text{sin}\\,\\beta $$<\/li>\n
  11. $$\\text{cos}(\\alpha \u00b1\\beta )=\\text{cos}\\,\\alpha \\,\\text{cos}\\,\\beta \\mp \\text{sin}\\,\\alpha \\,\\text{sin}\\,\\beta $$<\/li>\n
  12. $$\\text{tan}(\\alpha \u00b1\\beta )=\\frac{\\text{tan}\\,\\alpha \u00b1\\text{tan}\\,\\beta }{1\\mp \\text{tan}\\,\\alpha \\,\\text{tan}\\,\\beta }$$<\/li>\n
  13. $$\\text{sin}\\,2\\theta =2\\text{sin}\\,\\theta \\,\\text{cos}\\,\\theta $$<\/li>\n
  14. $$\\text{cos}\\,2\\theta ={\\text{cos}}^{2}\\,\\theta -{\\text{sin}}^{2}\\,\\theta =2\\,{\\text{cos}}^{2}\\,\\theta -1=1-2\\,{\\text{sin}}^{2}\\,\\theta $$<\/li>\n
  15. $$\\text{sin}\\,\\alpha +\\text{sin}\\,\\beta =2\\,\\text{sin}\\frac{1}{2}(\\alpha +\\beta )\\text{cos}\\frac{1}{2}(\\alpha -\\beta )$$<\/li>\n
  16. $$\\text{cos}\\,\\alpha +\\text{cos}\\,\\beta =2\\,\\text{cos}\\frac{1}{2}(\\alpha +\\beta )\\text{cos}\\frac{1}{2}(\\alpha -\\beta )$$<\/li>\n<\/ol>

    Triangles<\/em><\/p>\n

    1. Law of sines: $$ \\frac{a}{\\text{sin}\\,\\alpha }=\\frac{b}{\\text{sin}\\,\\beta }=\\frac{c}{\\text{sin}\\,\\gamma }$$<\/li>\n
    2. Law of cosines: $$ {c}^{2}={a}^{2}+{b}^{2}-2ab\\,\\text{cos}\\,\\gamma $$\n\n

      \"Figure<\/span><\/p><\/li>\n

    3. Pythagorean theorem: $$ {a}^{2}+{b}^{2}={c}^{2}$$\n\n

      \"Figure<\/span><\/p><\/li>\n<\/ol>

      Series expansions<\/strong><\/p>\n

      1. Binomial theorem: $$ {(a+b)}^{n}={a}^{n}+n{a}^{n-1}b+\\frac{n(n-1){a}^{n-2}{b}^{2}}{2\\text{!}}+\\frac{n(n-1)(n-2){a}^{n-3}{b}^{3}}{3\\text{!}}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
      2. $${(1\u00b1x)}^{n}=1\u00b1\\frac{nx}{1\\text{!}}+\\frac{n(n-1){x}^{2}}{2\\text{!}}\u00b1\\text{\u00b7\u00b7\u00b7}({x}^{2}<1)$$<\/li>\n
      3. $${(1\u00b1x)}^{\\text{\u2212}n}=1\\mp \\frac{nx}{1\\text{!}}+\\frac{n(n+1){x}^{2}}{2\\text{!}}\\mp \\text{\u00b7\u00b7\u00b7}({x}^{2}<1)$$<\/li>\n
      4. $$\\text{sin}\\,x=x-\\frac{{x}^{3}}{3\\text{!}}+\\frac{{x}^{5}}{5\\text{!}}-\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
      5. $$\\text{cos}\\,x=1-\\frac{{x}^{2}}{2\\text{!}}+\\frac{{x}^{4}}{4\\text{!}}-\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
      6. $$\\text{tan}\\,x=x+\\frac{{x}^{3}}{3}+\\frac{2{x}^{5}}{15}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
      7. $${e}^{x}=1+x+\\frac{{x}^{2}}{2\\text{!}}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
      8. $$\\text{ln}(1+x)=x-\\frac{1}{2}{x}^{2}+\\frac{1}{3}{x}^{3}-\\text{\u00b7\u00b7\u00b7}(|x|<1)$$<\/li>\n<\/ol>

        Derivatives<\/strong><\/p>\n

        1. $$\\frac{d}{dx}[af(x)]=a\\frac{d}{dx}f(x)$$<\/li>\n
        2. $$\\frac{d}{dx}[f(x)+g(x)]=\\frac{d}{dx}f(x)+\\frac{d}{dx}g(x)$$<\/li>\n
        3. $$\\frac{d}{dx}[f(x)g(x)]=f(x)\\frac{d}{dx}g(x)+g(x)\\frac{d}{dx}f(x)$$<\/li>\n
        4. $$\\frac{d}{dx}f(u)=[\\frac{d}{du}f(u)]\\frac{du}{dx}$$<\/li>\n
        5. $$\\frac{d}{dx}{x}^{m}=m{x}^{m-1}$$<\/li>\n
        6. $$\\frac{d}{dx}\\,\\text{sin}\\,x=\\text{cos}\\,x$$<\/li>\n
        7. $$\\frac{d}{dx}\\,\\text{cos}\\,x=\\text{\u2212}\\text{sin}\\,x$$<\/li>\n
        8. $$\\frac{d}{dx}\\,\\text{tan}\\,x={\\text{sec}}^{2}\\,x$$<\/li>\n
        9. $$\\frac{d}{dx}\\,\\text{cot}\\,x=\\text{\u2212}{\\text{csc}}^{2}\\,x$$<\/li>\n
        10. $$\\frac{d}{dx}\\,\\text{sec}\\,x=\\text{tan}\\,x\\,\\text{sec}\\,x$$<\/li>\n
        11. $$\\frac{d}{dx}\\,\\text{csc}\\,x=\\text{\u2212}\\text{cot}\\,x\\,\\text{csc}\\,x$$<\/li>\n
        12. $$\\frac{d}{dx}{e}^{x}={e}^{x}$$<\/li>\n
        13. $$\\frac{d}{dx}\\,\\text{ln}\\,x=\\frac{1}{x}$$<\/li>\n
        14. $$\\frac{d}{dx}\\,{\\text{sin}}^{-1}\\,x=\\frac{1}{\\sqrt{1-{x}^{2}}}$$<\/li>\n
        15. $$\\frac{d}{dx}\\,{\\text{cos}}^{-1}x=-\\frac{1}{\\sqrt{1-{x}^{2}}}$$<\/li>\n
        16. $$\\frac{d}{dx}\\,{\\text{tan}}^{-1}x=-\\frac{1}{1+{x}^{2}}$$<\/li>\n<\/ol>

          Integrals<\/strong><\/p>\n

          1. $$\\int af(x)dx=a\\int f(x)dx$$<\/li>\n
          2. $$\\int [f(x)+g(x)]dx=\\int f(x)dx+\\int g(x)dx$$<\/li>\n
          3. $$\\begin{array}{cc}\\hfill \\int {x}^{m}dx& =\\frac{{x}^{m+1}}{m+1}\\,(m\\ne \\text{\u2212}1)\\hfill \\\\ & =\\text{ln}\\,x(m=-1)\\hfill \\end{array}$$<\/li>\n
          4. $$\\int \\text{sin}\\,x\\,dx=\\text{\u2212}\\text{cos}\\,x$$<\/li>\n
          5. $$\\int \\text{cos}\\,x\\,dx=\\text{sin}\\,x$$<\/li>\n
          6. $$\\int \\text{tan}\\,x\\,dx=\\text{ln}|\\text{sec}\\,x|$$<\/li>\n
          7. $$\\int {\\text{sin}}^{2}\\,ax\\,dx=\\frac{x}{2}-\\frac{\\text{sin}\\,2ax}{4a}$$<\/li>\n
          8. $$\\int {\\text{cos}}^{2}\\,ax\\,dx=\\frac{x}{2}+\\frac{\\text{sin}\\,2ax}{4a}$$<\/li>\n
          9. $$\\int \\text{sin}\\,ax\\,\\text{cos}\\,ax\\,dx=-\\frac{\\text{cos}2ax}{4a}$$<\/li>\n
          10. $$\\int {e}^{ax}\\,dx=\\frac{1}{a}{e}^{ax}$$<\/li>\n
          11. $$\\int x{e}^{ax}dx=\\frac{{e}^{ax}}{{a}^{2}}(ax-1)$$<\/li>\n
          12. $$\\int \\text{ln}\\,ax\\,dx=x\\,\\text{ln}\\,ax-x$$<\/li>\n
          13. $$\\int \\frac{dx}{{a}^{2}+{x}^{2}}=\\frac{1}{a}\\,{\\text{tan}}^{-1}\\frac{x}{a}$$<\/li>\n
          14. $$\\int \\frac{dx}{{a}^{2}-{x}^{2}}=\\frac{1}{2a}\\,\\text{ln}|\\frac{x+a}{x-a}|$$<\/li>\n
          15. $$\\int \\frac{dx}{\\sqrt{{a}^{2}+{x}^{2}}}={\\text{sinh}}^{-1}\\frac{x}{a}$$<\/li>\n
          16. $$\\int \\frac{dx}{\\sqrt{{a}^{2}-{x}^{2}}}={\\text{sin}}^{-1}\\frac{x}{a}$$<\/li>\n
          17. $$\\int \\sqrt{{a}^{2}+{x}^{2}}\\,dx=\\frac{x}{2}\\sqrt{{a}^{2}+{x}^{2}}+\\frac{{a}^{2}}{2}\\,{\\text{sinh}}^{-1}\\frac{x}{a}$$<\/li>\n
          18. $$\\int \\sqrt{{a}^{2}-{x}^{2}}\\,dx=\\frac{x}{2}\\sqrt{{a}^{2}-{x}^{2}}+\\frac{{a}^{2}}{2}\\,{\\text{sin}}^{-1}\\frac{x}{a}$$<\/li>\n<\/ol>","rendered":"

            Quadratic formula<\/strong><\/p>\n

            If $$ a{x}^{2}+bx+c=0, $$ then $$ x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}$$<\/p>\n\n\n\n\n\n\n\n
            Geometry<\/span><\/caption>\n
            Triangle of base $$ b $$ and height $$ h$$<\/th>\nArea $$ =\\frac{1}{2}bh$$<\/th>\n\n<\/th>\n<\/tr>\n<\/thead>\n
            Circle of radius $$ r$$<\/td>\nCircumference $$ =2\\pi r$$<\/td>\nArea $$ =\\pi {r}^{2}$$<\/td>\n<\/tr>\n
            Sphere of radius $$ r$$<\/td>\nSurface area $$ =4\\pi {r}^{2}$$<\/td>\nVolume $$ =\\frac{4}{3}\\pi {r}^{3}$$<\/td>\n<\/tr>\n
            Cylinder of radius $$ r $$ and height $$ h$$<\/td>\nArea of curved surface $$ =2\\pi rh$$<\/td>\nVolume $$ =\\pi {r}^{2}h$$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            Trigonometry<\/strong><\/p>\n

            Trigonometric Identities<\/em><\/p>\n

              \n
            1. $$\\text{sin}\\,\\theta =1\\text{\/}\\text{csc}\\,\\theta $$<\/li>\n
            2. $$\\text{cos}\\,\\theta =1\\text{\/}\\text{sec}\\,\\theta $$<\/li>\n
            3. $$\\text{tan}\\,\\theta =1\\text{\/}\\text{cot}\\,\\theta $$<\/li>\n
            4. $$\\text{sin}({90}^{0}-\\theta )=\\text{cos}\\,\\theta $$<\/li>\n
            5. $$\\text{cos}({90}^{0}-\\theta )=\\text{sin}\\,\\theta $$<\/li>\n
            6. $$\\text{tan}({90}^{0}-\\theta )=\\text{cot}\\,\\theta $$<\/li>\n
            7. $${\\text{sin}}^{2}\\,\\theta +{\\text{cos}}^{2}\\,\\theta =1$$<\/li>\n
            8. $${\\text{sec}}^{2}\\,\\theta -{\\text{tan}}^{2}\\,\\theta =1$$<\/li>\n
            9. $$\\text{tan}\\,\\theta =\\text{sin}\\,\\theta \\text{\/}\\text{cos}\\,\\theta $$<\/li>\n
            10. $$\\text{sin}(\\alpha \u00b1\\beta )=\\text{sin}\\,\\alpha \\,\\text{cos}\\,\\beta \u00b1\\text{cos}\\,\\alpha \\,\\text{sin}\\,\\beta $$<\/li>\n
            11. $$\\text{cos}(\\alpha \u00b1\\beta )=\\text{cos}\\,\\alpha \\,\\text{cos}\\,\\beta \\mp \\text{sin}\\,\\alpha \\,\\text{sin}\\,\\beta $$<\/li>\n
            12. $$\\text{tan}(\\alpha \u00b1\\beta )=\\frac{\\text{tan}\\,\\alpha \u00b1\\text{tan}\\,\\beta }{1\\mp \\text{tan}\\,\\alpha \\,\\text{tan}\\,\\beta }$$<\/li>\n
            13. $$\\text{sin}\\,2\\theta =2\\text{sin}\\,\\theta \\,\\text{cos}\\,\\theta $$<\/li>\n
            14. $$\\text{cos}\\,2\\theta ={\\text{cos}}^{2}\\,\\theta -{\\text{sin}}^{2}\\,\\theta =2\\,{\\text{cos}}^{2}\\,\\theta -1=1-2\\,{\\text{sin}}^{2}\\,\\theta $$<\/li>\n
            15. $$\\text{sin}\\,\\alpha +\\text{sin}\\,\\beta =2\\,\\text{sin}\\frac{1}{2}(\\alpha +\\beta )\\text{cos}\\frac{1}{2}(\\alpha -\\beta )$$<\/li>\n
            16. $$\\text{cos}\\,\\alpha +\\text{cos}\\,\\beta =2\\,\\text{cos}\\frac{1}{2}(\\alpha +\\beta )\\text{cos}\\frac{1}{2}(\\alpha -\\beta )$$<\/li>\n<\/ol>\n

              Triangles<\/em><\/p>\n

                \n
              1. Law of sines: $$ \\frac{a}{\\text{sin}\\,\\alpha }=\\frac{b}{\\text{sin}\\,\\beta }=\\frac{c}{\\text{sin}\\,\\gamma }$$<\/li>\n
              2. Law of cosines: $$ {c}^{2}={a}^{2}+{b}^{2}-2ab\\,\\text{cos}\\,\\gamma $$\n

                \"Figure<\/span><\/p>\n<\/li>\n

              3. Pythagorean theorem: $$ {a}^{2}+{b}^{2}={c}^{2}$$\n

                \"Figure<\/span><\/p>\n<\/li>\n<\/ol>\n

                Series expansions<\/strong><\/p>\n

                  \n
                1. Binomial theorem: $$ {(a+b)}^{n}={a}^{n}+n{a}^{n-1}b+\\frac{n(n-1){a}^{n-2}{b}^{2}}{2\\text{!}}+\\frac{n(n-1)(n-2){a}^{n-3}{b}^{3}}{3\\text{!}}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
                2. $${(1\u00b1x)}^{n}=1\u00b1\\frac{nx}{1\\text{!}}+\\frac{n(n-1){x}^{2}}{2\\text{!}}\u00b1\\text{\u00b7\u00b7\u00b7}({x}^{2}<1)$$<\/li>\n
                3. $${(1\u00b1x)}^{\\text{\u2212}n}=1\\mp \\frac{nx}{1\\text{!}}+\\frac{n(n+1){x}^{2}}{2\\text{!}}\\mp \\text{\u00b7\u00b7\u00b7}({x}^{2}<1)$$<\/li>\n
                4. $$\\text{sin}\\,x=x-\\frac{{x}^{3}}{3\\text{!}}+\\frac{{x}^{5}}{5\\text{!}}-\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
                5. $$\\text{cos}\\,x=1-\\frac{{x}^{2}}{2\\text{!}}+\\frac{{x}^{4}}{4\\text{!}}-\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
                6. $$\\text{tan}\\,x=x+\\frac{{x}^{3}}{3}+\\frac{2{x}^{5}}{15}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
                7. $${e}^{x}=1+x+\\frac{{x}^{2}}{2\\text{!}}+\\text{\u00b7\u00b7\u00b7}$$<\/li>\n
                8. $$\\text{ln}(1+x)=x-\\frac{1}{2}{x}^{2}+\\frac{1}{3}{x}^{3}-\\text{\u00b7\u00b7\u00b7}(|x|<1)$$<\/li>\n<\/ol>\n

                  Derivatives<\/strong><\/p>\n

                    \n
                  1. $$\\frac{d}{dx}[af(x)]=a\\frac{d}{dx}f(x)$$<\/li>\n
                  2. $$\\frac{d}{dx}[f(x)+g(x)]=\\frac{d}{dx}f(x)+\\frac{d}{dx}g(x)$$<\/li>\n
                  3. $$\\frac{d}{dx}[f(x)g(x)]=f(x)\\frac{d}{dx}g(x)+g(x)\\frac{d}{dx}f(x)$$<\/li>\n
                  4. $$\\frac{d}{dx}f(u)=[\\frac{d}{du}f(u)]\\frac{du}{dx}$$<\/li>\n
                  5. $$\\frac{d}{dx}{x}^{m}=m{x}^{m-1}$$<\/li>\n
                  6. $$\\frac{d}{dx}\\,\\text{sin}\\,x=\\text{cos}\\,x$$<\/li>\n
                  7. $$\\frac{d}{dx}\\,\\text{cos}\\,x=\\text{\u2212}\\text{sin}\\,x$$<\/li>\n
                  8. $$\\frac{d}{dx}\\,\\text{tan}\\,x={\\text{sec}}^{2}\\,x$$<\/li>\n
                  9. $$\\frac{d}{dx}\\,\\text{cot}\\,x=\\text{\u2212}{\\text{csc}}^{2}\\,x$$<\/li>\n
                  10. $$\\frac{d}{dx}\\,\\text{sec}\\,x=\\text{tan}\\,x\\,\\text{sec}\\,x$$<\/li>\n
                  11. $$\\frac{d}{dx}\\,\\text{csc}\\,x=\\text{\u2212}\\text{cot}\\,x\\,\\text{csc}\\,x$$<\/li>\n
                  12. $$\\frac{d}{dx}{e}^{x}={e}^{x}$$<\/li>\n
                  13. $$\\frac{d}{dx}\\,\\text{ln}\\,x=\\frac{1}{x}$$<\/li>\n
                  14. $$\\frac{d}{dx}\\,{\\text{sin}}^{-1}\\,x=\\frac{1}{\\sqrt{1-{x}^{2}}}$$<\/li>\n
                  15. $$\\frac{d}{dx}\\,{\\text{cos}}^{-1}x=-\\frac{1}{\\sqrt{1-{x}^{2}}}$$<\/li>\n
                  16. $$\\frac{d}{dx}\\,{\\text{tan}}^{-1}x=-\\frac{1}{1+{x}^{2}}$$<\/li>\n<\/ol>\n

                    Integrals<\/strong><\/p>\n

                      \n
                    1. $$\\int af(x)dx=a\\int f(x)dx$$<\/li>\n
                    2. $$\\int [f(x)+g(x)]dx=\\int f(x)dx+\\int g(x)dx$$<\/li>\n
                    3. $$\\begin{array}{cc}\\hfill \\int {x}^{m}dx& =\\frac{{x}^{m+1}}{m+1}\\,(m\\ne \\text{\u2212}1)\\hfill \\\\ & =\\text{ln}\\,x(m=-1)\\hfill \\end{array}$$<\/li>\n
                    4. $$\\int \\text{sin}\\,x\\,dx=\\text{\u2212}\\text{cos}\\,x$$<\/li>\n
                    5. $$\\int \\text{cos}\\,x\\,dx=\\text{sin}\\,x$$<\/li>\n
                    6. $$\\int \\text{tan}\\,x\\,dx=\\text{ln}|\\text{sec}\\,x|$$<\/li>\n
                    7. $$\\int {\\text{sin}}^{2}\\,ax\\,dx=\\frac{x}{2}-\\frac{\\text{sin}\\,2ax}{4a}$$<\/li>\n
                    8. $$\\int {\\text{cos}}^{2}\\,ax\\,dx=\\frac{x}{2}+\\frac{\\text{sin}\\,2ax}{4a}$$<\/li>\n
                    9. $$\\int \\text{sin}\\,ax\\,\\text{cos}\\,ax\\,dx=-\\frac{\\text{cos}2ax}{4a}$$<\/li>\n
                    10. $$\\int {e}^{ax}\\,dx=\\frac{1}{a}{e}^{ax}$$<\/li>\n
                    11. $$\\int x{e}^{ax}dx=\\frac{{e}^{ax}}{{a}^{2}}(ax-1)$$<\/li>\n
                    12. $$\\int \\text{ln}\\,ax\\,dx=x\\,\\text{ln}\\,ax-x$$<\/li>\n
                    13. $$\\int \\frac{dx}{{a}^{2}+{x}^{2}}=\\frac{1}{a}\\,{\\text{tan}}^{-1}\\frac{x}{a}$$<\/li>\n
                    14. $$\\int \\frac{dx}{{a}^{2}-{x}^{2}}=\\frac{1}{2a}\\,\\text{ln}|\\frac{x+a}{x-a}|$$<\/li>\n
                    15. $$\\int \\frac{dx}{\\sqrt{{a}^{2}+{x}^{2}}}={\\text{sinh}}^{-1}\\frac{x}{a}$$<\/li>\n
                    16. $$\\int \\frac{dx}{\\sqrt{{a}^{2}-{x}^{2}}}={\\text{sin}}^{-1}\\frac{x}{a}$$<\/li>\n
                    17. $$\\int \\sqrt{{a}^{2}+{x}^{2}}\\,dx=\\frac{x}{2}\\sqrt{{a}^{2}+{x}^{2}}+\\frac{{a}^{2}}{2}\\,{\\text{sinh}}^{-1}\\frac{x}{a}$$<\/li>\n
                    18. $$\\int \\sqrt{{a}^{2}-{x}^{2}}\\,dx=\\frac{x}{2}\\sqrt{{a}^{2}-{x}^{2}}+\\frac{{a}^{2}}{2}\\,{\\text{sin}}^{-1}\\frac{x}{a}$$<\/li>\n<\/ol>\n\n\t\t\t
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